Stationary solutions of SPDEs and infinite horizon BDSDEs

Abstract

In this paper we study the existence of stationary solutions for stochastic partial differential equations. We establish a new connection between L 2 ρ (R d ; R 1 ) ⊗ L 2 ρ (R d ; R d ) valued solutions of backward doubly stochastic differential equations (BDSDEs) on infinite horizon and the stationary solutions of the SPDEs. Moreover, we prove the existence and uniqueness of the solutions of BDSDEs on both finite and infinite horizons, so obtain the solutions of initial value problems and the stationary solutions (independent of any initial value) of SPDEs. The connection of the weak solutions of SPDEs and BDSDEs has independent interests in the areas of both SPDEs and BSDEs.

Pathwise Properties of Random Quadratic Mapping

“…In the last thirty years, the research of random dynamical systems is further expanded especially in the field of stochastic differential equations and stochastic partial differential equations in a series of work such as [4,23,24,28,31]. Pathwise stationary solution and random periodic solution are two central concepts in the study of random dynamical systems [4,13,14,21,20,29,31,36,37,42,43,44]. To study them is key towards understanding the longtime behavior of the random dynamical systems and their local and global topological structure.…”

Anticipating random periodic solutions—I. SDEs with multiplicative linear noise

“…In general, 1 arXiv:1502.00567v2 [math.PR] 27 Apr 2016 random perturbations to a periodic solution break the strict periodicity immediately, similar to the case that random perturbations break fixed points. The concept of stationary solutions is the corresponding notion of fixed points in the stochastic counterpart and has been subject to intensive study in mathematics literature ( [11], [21], [27], [32]). One of the obstacles to make a systematic progress in the study of the random periodicity was the lack of a rigorous mathematical definition in a great generality and appropriate mathematical tools.…”

Probabilistic Representation of Weak Solutions of Partial Differential Equations with Polynomial Growth Coefficients

“…The solution of the BSDEs in above cases gives the probabilistic representation of the classical or viscosity solution of the PDEs as a generalization to the Feynman-Kac formula. Applications of BSDEs have been found in some problems such as a model in mathematics of finance (El Karoui, Peng and Quenez [11]), as an efficient method for constructing Γ -martingales on Riemannian manifolds (Darling [5]), and as an intrinsic tool to construct the pathwise stationary solution for stochastic PDEs (Zhang and Zhao [25], [26]). The Feynman-Kac approach to a Sobolev or L 2 space valued weak solution of PDEs has been concentrated mainly on linear problems.…”

“…It is also inadequate to use a combination of the weak convergence in finite dimensional space developed by Pardoux [18] and the weak solution method developed by Bally and Matoussi [1], Zhang and Zhao [25], [26], to solve this problem. We need to introduce some new ideas to the study of BSDEs.…”